Question 35
For a positive integer $N$, we color the positive divisors of $N$ (including $1$ and $N$) with four colors. A coloring is called multichromatic if whenever $a$, $b$ and $\operatorname{gcd}(a, b)$ are pairwise distinct divisors of $N$, then they have pairwise distinct colors. What is the maximum possible number of multichromatic colorings a positive integer can have if it is not the power of any prime?